Semiparametric Approach for Regression with Covariate Subject to Limit of Detection

We consider generalized linear regression analysis with left-censored covariate due to the lower limit of detection. Complete case analysis by eliminating observations with values below limit of detection yields valid estimates for regression coefficients, but loses efficiency; substitution methods are biased; maximum likelihood method relies on parametric models for the unobservable tail probability distribution of such covariate, thus may suffer from model misspecification. To obtain robust and more efficient results, we propose a semiparametric likelihood-based approach for the estimation of regression parameters using an accelerated failure time model for the covariate subject to limit of detection. A two-stage estimation procedure is considered, where the conditional distribution of the covariate with limit of detection given other variables is estimated prior to maximizing the likelihood function. The proposed method outperforms the complete case analysis and the substitution methods as well in simulation studies. Technical conditions for desirable asymptotic properties are provided.